- #1
issacnewton
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- 36
Homework Statement
Verify that [itex]y_1=x^3[/itex] and [itex]y_2=|x|^3[/itex] are linearly independent
solutions of the differential equation [itex]x^2y''-4xy'+6y=0 [/itex] on the interval
[itex](-\infty,\infty)[/itex]. Show that [itex]W(y_1,y_2)=0[/itex] for every real number
x, where W is the wronskian.
Homework Equations
theorems on differential equations
The Attempt at a Solution
First I need to check that y1 and y2 are the solutions of the
given diff. equation. y1 is easy. To prove that y2 is the solution,
I divided the whole interval [itex](-\infty,\infty)[/itex], in three parts [itex]x>0\; ,x=0\;,\;x>0[/itex]. And then I showed that the diff. equation is satisfied on all the different
parts. So that means , y2 is the solution of the diff. equation
Now, to check the linear independence, let's consider the equation
[tex]c_1 x^3+c_2 |x|^3=0[/tex]
Now here I am stuck. How do I prove that [itex]c_1=c_2=0[/itex] for all values of x in
[itex](-\infty,\infty)[/itex].
thanks